Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic ModelsCurve Fitting with Exponential and Logarithmic Models

Holt Algebra 2Holt McDougal Algebra 2

• How do we model data by using exponential and logarithmic functions?

• How do we use exponential and logarithmic models to analyze and predict?

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

In previous chapters, you used a graphing calculator to perform linear progressions and quadratic regressions to make predictions. You can also use an exponential model, which is an exponential function that represents a real data set.

Once you know that data are exponential, you can use ExpReg (exponential regression) on your calculator to find a function that fits. This method of using data to find an exponential model is called an exponential regression. The calculator fits exponential functions to abx, so translations cannot be modeled.

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

If you do not see r2 and r when you calculate regression, and turn these on by selecting DiagnosticOn.

Remember!

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

1. Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000.

College Application

Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature.

Tuition of the University of Texas

Year Tuition

1999–00 $3128

2000–01 $3585

2001–02 $3776

2002–03 $3950

2003–04 $4188

An exponential model is f(x) ≈ 3236(1.07x), where f(x) represents the tuition and x is the number of years after the 1999–2000 year.

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

1. Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000.

College Application

Step 2 Graph the data and the function model to verify that it fits the data.

Tuition of the University of Texas

Year Tuition

1999–00 $3128

2000–01 $3585

2001–02 $3776

2002–03 $3950

2003–04 $4188Graph the line y = 6000 on Y2.

Zoom Out, then use 2nd Trace (Calc) and hit Intersect

7500

1500

The tuition will be about $6000 when t = 9 or 2008–09.

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

2. Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000?

College Application

Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature.

An exponential model is f(x) ≈ 199(1.25x), where f(x) represents the number of bacteria and x is the number of minutes.

Time (min) 0 1 2 3 4 5

Bacteria 200 248 312 390 489 610

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

2. Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000?

College Application

Time (min) 0 1 2 3 4 5

Bacteria 200 248 312 390 489 610

Step 2 Graph the data and the function model to verify that it fits the data.

Graph the line y = 2000 on Y2.

Zoom Out, then use 2nd Trace (Calc) and hit Intersect

The bacteria will be about 2000 when x = 10.3 min.

2500

00 15

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

Many natural phenomena can be modeled by natural log functions. You can use a logarithmic regression to find a function

Most calculators that perform logarithmic regression use ln rather than log.

Helpful Hint

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

3. Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000?

Application

Step 1 Enter data into two lists in a graphing calculator. Use the logarithmic regression feature.

An logarithmic model is f(x) ≈ 1824 + 106ln x, where f(x) is the year and x is the population in billions.

Global Population Growth

Population (billions)

Year

1 1800

2 1927

3 1960

4 1974

5 1987

6 1999

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

Application

Step 2 Graph the data and the function model to verify that it fits the data.

Because we are looking for the y-value when x = 9, you can use the table and scroll down to x = 9.

The population will exceed 9,000,000,000 in the year 2057.

Global Population Growth

Population (billions)

Year

1 1800

2 1927

3 1960

4 1974

5 1987

6 1999

0

2500

0 15

3. Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000?

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

4. Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s?

Application

Step 1 Enter data into two lists in a graphing calculator. Use the logarithmic regression feature.

An logarithmic model is f(x) ≈ 0.59 + 2.64 ln x, where f(x) is the time and x is the speed.

Time (min) 1 2 3 4 5 6 7

Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

Application

Step 2 Graph the data and the function model to verify that it fits the data.

Graph the line y = 8 on Y2.

Zoom Out, then use 2nd Trace (Calc) and hit Intersect

The time it will reach 8.0 m/sec is when x = 16.6 min.

4. Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s?

Time (min) 1 2 3 4 5 6 7

Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6

0

10

200

Holt McDougal Algebra 2

Curve Fitting with Exponentialand Logarithmic Models

Lesson 15.2 Practice B